Inelastic Neutron Scattering

by equation (2.11) [23, 24].

R_p= ^∑i|y_i,obs−y_i,calc|

∑iy_i,obs and R_wp=

s∑iw_i· |y_i,obs−y_i,calc|²

∑iy_i,obs (2.11)

In equation (2.11), y_i,obs/calc are the intensities of measured ("obs") and calculated ("calc") profile points andw_i is a weighting factor.

In general, all Rvalues are given in percentages and in an ideal case they become zero as in that case the error sum of squares becomes zero [23, 24, 22].

Finally, theχ²value is a statistical parameter which is used in the evaluation of sin-gle crystal and powder refinements. This parameter is normalized to the statistical degrees of freedom, defined by the difference between the number of observed data points, N_obs, and the number of parameters, N_parameter, involved in the refinement step (equation (2.12)).

χ² = ^∑iwi· |δ|²

N_obs−N_parameter (2.12)

Here,δis the difference between the measured and calculated intensities either from a single crystal or powder experiment (see also equation (2.4)). In an ideal caseχ² becomes 1 [23, 24, 22].

2.2 Inelastic Neutron Scattering

2.2.1 Elastic and inelastic scattering

If a neutron beam interacts with a sample different scattering processes which are -in general - the elastic and -inelastic scatter-ing, occur -in parallel (figure 2.1). In the following figure the wave vector of the incident beam is denoted ask₀, while the one of the scattered beam corresponds tok_s.

FIGURE2.1: The scattering triangles for elastic (black) and inelastic scattering with phonon creation (red) and phonon annihilation (blue),

respectively.

In case of an elastic scattering process, the energy,E₀, of the incident beam is equal to

14 Chapter 2. Theory the one of the scattered beam,E_s, and only the direction ofk_sis changed with respect to the one ofk₀[28, 29]. The difference vector,Q, betweenk_sandk₀corresponds to a reciprocal lattice vector,G, so that for an elastic scattering processk_s−k₀ = Q= G (figure 2.1).

In the sample, phonons with discrete energy levels, ¯h·ω (¯h: reduced Planck con-stant; ω: Phonon frequency), are present. In an inelastic scattering process, the in-cident beam interacts under an energy (¯h·ω) and momentum (¯h·q) transfer with the phonons of the sample [28, 29]. This interaction results either in a creation or an annihilation of a phonon in the sample [28, 29]. Equations (2.13a) and (2.13b) sum-marize the underlying law of energy and momentum conversion, respectively [28, 29].

E_s−E₀ =±¯hω (2.13a)

¯

hk_s−¯hk₀= ¯hQ=hG¯ ±¯hq (2.13b) The phonon creation and annihilation, respectively, is taken into account by the± sign. While (+) corresponds to an annihilation process, (−) describes the phonon creation [28, 29]. Here, ¯hq corresponds to the so-called crystal momentum, which describes the momentum transferred to the sample [28, 29]. Figure 2.1 compares the scattering triangles for the phonon creation and annihilation process with the one of the elastic scattering. The contribution of the phonon wave vectorqis reflected by the length of thek_svector.

2.2.2 Scattering cross section and scattering function

In a neutron scattering experiment one measures the fraction of the flux Φof the incident neutron beam scattered by the sample [30]. This fraction is denoted as scat-tering cross sectionσ. In case one measures the total signal scattered by the sample, one measures the total scattering cross section [30]. If only the part of the neutron beam scattered in a solid angular elementdΩis measured, one determines the dif-ferential cross section_dΩ^dσ [30]. The differential cross section corresponds to an elastic scattering experiment as the energy of the scattered neutrons remains unchanged [30]. In case one measures the neutron beam scattered by a sample indΩand in an energy interval dE, one determines the double differential scattering cross section

d²σ

dΩdE [30]. The double differential cross section describes an inelastic scattering ex-periment as it takes into account that the energy of the scattered neutrons is changed due to the scattering process in the sample [30].

The double differential cross section is connected to the scattering function,S(Q,E) (equation (2.14)), wherek₀andk_srepresent the wave vector of the incident and scat-tered beam, respectively [31].

d²σ dEdΩ = ^ks

k₀ ·S(Q,E) (2.14)

2.2. Inelastic Neutron Scattering 15 The scattering function describes all kinds of interactions between the incident neu-tron beam and the sample. It was demonstrated [32, 33] that in terms of the conven-tional harmonic phonon expansion the scattering function,S(Q,E), can be expressed by a sum of an elastic (S₀(Q,E)), an one-phonon (S₁(Q,E)) and a multi-phonon (S_m(Q,E)) contribution (equation (2.15)).

S(Q,E) =S₀(Q,E) +S₁(Q,E) +S_m(Q,E) (2.15) Each of these scattering functions has a coherent and an incoherent contribution [30, 32, 33]. In case of coherent scattering, there are fixed phase relations between the scattered waves and thus, constructive interference occurs. Moreover, a coherent scattering process depends on the direction of the scattering vector. In contrast to this, in an incoherent scattering process the phase relations between the scattered waves are random and thus, a constructive interference is not possible. Moreover, the incoherent scattering is uniform and isotropic in all directions [30].

Bragg peaks are the result of a coherent elastic scattering process as the periodicity of the sample’s crystal structure allows a constructive interference between the scat-tered waves [30, 32, 33]. Incoherent elastic scattering reflects single-particle motion and thus, contributes only to the background in a diffraction experiment [30].

2.2.3 The one-phonon scattering function and the incoherent approxima-tion

For an inelastic neutron scattering from a polycrystalline sample the one-phonon scattering function, S₁(Q,E), on the neutron-energy loss side (Phonon creation) is described by the expression shown in equation (2.16) [33, 34, 32].

S₁(Q,E) = ¹

In equation (2.16), N, iand j, andω_m, correspond to the number of oscillators, the atomic speciesiandj, and the frequency of themth normal mode. The parameters b_i/j, W_i/j, R_i/j andξ_i/j are the neutron scattering lengthb, the Debye Waller factor W, the atomic equilibrium position R, and the displacement vector ξ, of i and j.

nm = [exp(¯hωm/k_BT)−1]⁻¹ is a population factor and the factor(Q·ξ_i)·(Q·ξ_j) describes the correlation between the atomic speciesiandj[33, 34, 32]. In general, equation (2.16) contains a coherent (i 6= j) and an incoherent (i = j) contribution, where the former one is usually the dominating one as the coherent scattering length of most isotopes is larger compared to the corresponding incoherent one.

Usually, the coherent inelastic neutron scattering from a polycrystalline sample is treated under the incoherent approximation [33, 34, 32]. In a polycrystalline sample the grains are randomly distributed and thus, the scattering functionS₁(Q,E)_{can be}

16 Chapter 2. Theory averaged over an extendedQrange [33, 34, 32]. The consequence of this is, that the correlations between the atomsiandjget lost, and hence, the factor(Q·ξ_i)·(Q·ξ_j) can be replaced by its average ¹₃·Q²·ξ²_iδ_ij [33, 34, 32]. As a result of the incoherent approximation the double differential cross sections of the coherent and incoherent scattering,(_dEd^d2σ_Ω)_coh≈(_dEd^d2σ_Ω)_incoh, are approximately identical [30].

In the scope of the incoherent approximation, the scattering functionS₁(Q,E)can be simplified to the expression shown in equation (2.17) [32, 34].

S₁(Q,E) =

∑

Obviously, the scattering functionS₁(Q,E)is connected with the sum of the partial phonon density of states, g_i(ω), of the atomsiin the sample. The partial phonon density of states are connected with the total one,G(ω), by equation (2.18).

G(ω) =

∑

Here, ^4π_m^·b2i

i is an element-specific weighting factor [33]. As one measures the total double differential cross section in an inelastic neutron scattering experiment, this technique gives access to the total phonon density of states,G(ω), of a compound.

2.2.4 The ARCS beamline

All inelastic neutron scattering experiments described in this study have been per-formed at the beamline ARCS at the Spallation Neutron Source (SNS) facility at the Oak Ridge National Laboratory (ORNL), Tennessee, USA. The beamline is well de-scribed in previous publications [35, 36] and thus, only a short description of the main parts is given here.

ARCS is a wide-angular time-of-flight chopper spectrometer. A scheme of the beam-line is shown in figure 2.2. At the T₀chopper, a polychromatic, short neutron pulse containing the desired energy range is selected by adjusting a suitable rotation speed.

Moreover, undesirable radiations (e. g.γ-rays) and too fast or too slow neutrons, re-spectively, are blocked [35, 36]. The selected neutron pulse is guided to the Fermi chopper where it is monochromatized. The initial intensity,I₀, of the monochrom-atized beam is detected by a beam monitor. The monochrommonochrom-atized beam is guided to the sample stage of the spectrometer which is equipped with a He cryostat. The flight-time between the Fermi chopper and the sample is well known. The monochro-matized neutron beam interacts with the sample and several scattering processes oc-cur. The scattered neutrons are detected and normalized toI₀later by the beamline software [35, 36]. While elastically scattered neutrons arrive at the detector array at the timet₀, inelastically scattered ones arrive either earlier (phonon annihilation, Anti-Stokes process) or later (phonon creation, Stokes process) [37].

The detector array of the beamline consists of 920 sensitive³He tubes and covers a